This document is a project report submitted by Tulasi Chandan Behera to the Department of Electrical Engineering at IIT Patna in partial fulfillment of the requirements for a Bachelor of Technology degree. The project involved numerical modelling of few mode fiber under the guidance of Dr. Sumanta Gupta. The report provides background on fiber optics, types of optical fibers, fiber impairments, basic fiber optic communication systems, modelling and implementation details using the split-step Fourier method, and results from modelling nonlinearity and space division multiplexing in few mode fibers.

1. NUMERICAL MODELLING OF FEW MODE FIBER
A B. Tech Project Report Submitted
in Partial Fulfillment of the Requirements
for the Degree of
Bachelor of Technology
by
Tulasi Chandan Behera
(1301EE42)
under the guidance of
Dr. Sumanta Gupta
to the
DEPARTMENT OF ELECTRICAL ENGINEERING
INDIAN INSTITUTE OF TECHNOLOGY PATNA
PATNA - 801106, BIHAR

2. 2

3. CERTIFICATE
This is to certify that the work contained in this thesis entitled “NUMERICAL MOD-
ELLING OF FEW MODE FIBER” is a bonafide work of Tulasi Chandan Behera
(Roll No. 1301EE42), carried out in the Department of Electrical Engineering, Indian
Institute of Technology Patna under my supervision and that it has not been submitted
elsewhere for a degree.
Supervisor: Dr. Sumanta Gupta
Associate Professor,
May, 2017 Department of Electrical & Engineering,
Patna. Indian Institute of Technology Patna, Bihar.
i

4. ii

5. Acknowledgements
I would like to express my gratitude towards all the people who have contributed their
precious time and effort to help me in the project. First and foremost, I would thank
Almighty God and my parents without whom I would not be what I am today. I would
like to thank Dr. Sumanta Gupta, Department of Electrical Engineering, IIT Patna, my
project supervisor for his guidance, support, motivation and encouragement throughout the
period this work was carried out. His readiness for consultation at all times, his educative
comments, his concern and assistance have been invaluable. Last but not least, I would also
extend my sincere thanks to Department of electrical engineering IIT Patna for availing
the facilities required for my investigation analysis.
iii

6. iv

7. Contents
List of Figures vii
1 Introduction 1
1.1 Benefits of fiber optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Long-distance signal transmission . . . . . . . . . . . . . . . . . . . 2
1.1.2 Large bandwidth, light weight, and small diameter . . . . . . . . . 2
1.1.3 Non-conductivity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.4 Security . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.1.5 Designed for future applications needs . . . . . . . . . . . . . . . . 3
1.2 Types of Optical Fiber: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2.1 According to the no of modes: . . . . . . . . . . . . . . . . . . . . . 3
1.2.2 According to the refractive index profile optical fibers: . . . . . . . 3
2 Fiber impairments: 5
2.1 Linear Impairments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 Attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.2 Chromatic Dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.3 Polarization mode dispersion . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Nonlinear Impairments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.1 Stimulated Raman Scattering (SRS) . . . . . . . . . . . . . . . . . 7
2.2.2 Stimulated Brillouin Scattering (SBS) . . . . . . . . . . . . . . . . . 8
v

8. 2.2.3 Four-wave mixing (FWM) . . . . . . . . . . . . . . . . . . . . . . . 8
2.2.4 Self- phase modulation (SPM) . . . . . . . . . . . . . . . . . . . . . 9
2.2.5 Cross phase modulation (XPM) . . . . . . . . . . . . . . . . . . . . 9
3 Basic fiber optic communication system 11
4 Modelling and implementation details 13
4.1 Split-step Fourier (SSF) method . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 Modelling of non-linearity in few mode fiber . . . . . . . . . . . . . . . . . 15
4.2.1 Modelling of Kerr nonlinear effects . . . . . . . . . . . . . . . . . . 15
4.3 Space division multiplexing in few mode fiber . . . . . . . . . . . . . . . . 17
4.4 MIMO signal processing in few mode fiber for SDM . . . . . . . . . . . . . 18
5 Results 19
6 Conclusion and discussion 25
References 29
vi

9. List of Figures
3.1 Basic fiber optic communication system . . . . . . . . . . . . . . . . . . . . 11
4.1 (1)Matrix model for an MMF system described by SSF method in cascade.(2)
Each matrix in the cascade may represent a section of MMF, a modal mul-
tiplexer or demultiplexer, a multimode optical amplifier or other components. 15
4.2 Matrix model for an MMF system described by the product of a cascade of
matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.3 Space division multiplexing(SDM) Using few mode fiber with DSP MIMO
processing unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.1 Scatter plot corresponding to mode1 in a two mode system when input power
=0dbm for both the signal entering into different modes not considering
nonlinearity in FMF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
5.2 Scatter plot corresponding to mode1 in a two mode system when input
power =0dbm for both the signal entering into different modes considering
nonlinearity in FMF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
5.3 Scatter plot corresponding to mode2 in a two mode system when input power
=0dbm for both the signal entering into different modes and not considering
nonlinearity in FMF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
vii

10. 5.4 Scatter plot corresponding to mode2 in a two mode system when input power
=0dbm for both the signal entering into different modes and considering
nonlinearity in FMF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.5 Scatter plot corresponding to mode1 in the first plot and mode2 in the
second plot in a two mode system when peak input power =-5dbm for both
the signal entering into different modes and not considering nonlinearity in
FMF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
5.6 Scatter plot corresponding to mode1 in the first plot and mode2 in the
second plot in a two mode system when peak input power =-5dbm for both
the signal entering into different modes and considering nonlinearity in FMF. 22
5.7 Scatter plot corresponding to mode1 in the first plot and mode2 in the
second plot in a two mode system when peak input power = 5dbm for both
the signal entering into different modes and not considering nonlinearity in
FMF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
5.8 Scatter plot corresponding to mode1 in the first plot and mode2 in the
second plot in a two mode system when peak input power = 5dbm for both
the signal entering into different modes and considering nonlinearity in FMF 23
5.9 Scatter plot corresponding to mode1 in the first plot and mode2 in the second
plot in a two mode system when peak input power for mode1= 7dbm for
mode1 and peak input power for mode2= 3dbm for the signal entering into
different modes and not considering nonlinearity in FMF. . . . . . . . . . . 23
5.10 Scatter plot corresponding to mode1 in the first plot and mode2 in the second
plot in a two mode system when peak input power for mode1= 7dbm for
mode1 and peak input power for mode2= 3dbm for the signal entering into
different modes and considering nonlinearity in FMF . . . . . . . . . . . . 24
viii

11. Chapter 1
Introduction
Fiber optics is a major building block in the communication infrastructure. Its high
bandwidth capabilities and low attenuation characteristics make it ideal for gigabit trans-
mission and beyond. Since its invention in the early 1970s, the use of and demand for optical
fiber have grown tremendously. The uses of optical fiber today are quite numerous. With
the explosion of information traffic due to the Internet, electronic commerce, computer
networks, multimedia, voice, data, and video, the need for a transmission medium with
the bandwidth capabilities for handling such vast amounts of information is paramount.
Fiber optics, with its comparatively infinite bandwidth, has proven to be the solution. The
growth of the fiber optics industry over the past five years has been explosive. Analysts
expect that this industry will continue to grow at a tremendous rate well into the next
decade and beyond. Anyone with a vested interest in communication field would be all the
wiser to learn more about the tremendous advantages of fiber optic communication.
1.1 Benefits of fiber optics
Optical fiber systems have many advantages over metallic-based communication systems.
These advantages include:
1

12. 1.1.1 Long-distance signal transmission
The low attenuation and superior signal integrity found in optical systems allow much
longer intervals of signal transmission than metallic-based systems.
1.1.2 Large bandwidth, light weight, and small diameter
Todays applications require an ever-increasing amount of bandwidth. Consequently, it is
important to consider the space constraints of many end users. It is commonplace to install
new cabling within existing duct systems or conduit. The relatively small diameter and
light weight of optical cable make such installations easy and practical, saving valuable
conduit space in these environments.
1.1.3 Non-conductivity
Another advantage of optical fibers is their dielectric nature. Since optical fiber has no
metallic components, it can be installed in areas with electromagnetic interference (EMI),
including radio frequency interference (RFI). Areas with high EMI include utility lines,
power-carrying lines, and railroad tracks. All-dielectric cables are also ideal for areas of
high lightning-strike incidence.
1.1.4 Security
Unlike metallic-based systems, the dielectric nature of optical fiber makes it impossible
to remotely detect the signal being transmitted within the cable. The only way to do so
is by accessing the optical fiber. Accessing the fiber requires intervention that is easily
detectable by security surveillance. These circumstances make fiber extremely attractive
to governmental bodies, banks, and others with major security concerns.
2

13. 1.1.5 Designed for future applications needs
Fiber optics is affordable today, as electronics prices fall and optical cable pricing remains
low. In many cases, fiber solutions are less costly than copper. As bandwidth demands
increase rapidly with technological advances, fiber will continue to play a vital role in the
long-term success of telecommunication.
1.2 Types of Optical Fiber:
1.2.1 According to the no of modes:
• Single Mode Fiber:
Single mode fibers are used to transmit one signal per fiber.The core diameter is
almost equal to the wave length of the emitted light so that it propagates along a
single path.
• Multi Mode Fiber:
In optical fibre technology, multimode fibre is optical fibre that is designed to carry
more than one signal at a time. Each carries multiple rays or modes concurrently,
each at a slightly different reflection angle with the optical fibre core.
1.2.2 According to the refractive index profile optical fibers:
• Step Index Fiber:
Core and Cladding material has uniform but different refractive index.
• Graded Index Fiber:
Core material has variable index as a function of the radial distance from the cen-
ter.The refractive index diminishes gradually from the center axis out toward the
cladding.
3

14. 4

15. Chapter 2
Fiber impairments:
2.1 Linear impairments
2.1.1 Attenuation
Attenuation causes the decay of signal strength, loss of light power as the signal propagates
through the fiber. Attenuation in optical fibers are caused by intrinsic factors which are
scattering , and absorption and by extrinsic factors which include stress from the manu-
facturing process, environmental and physical bending.
2.1.2 Chromatic Dispersion
Light pulses representing data have a definite spectral width. Because of chromatic disper-
sion in an optical fiber different wavelengths propagate at different speeds thereby resulting
pulse spreading. If left unmanaged, pulse spreading eventually results in inter-symbol in-
terference when adjacent pulses overlap leading to errors in the recovery of transmitted
bits.
5

16. Group velocity dispersion:
Group velocity dispersion (GVD) is the main cause for pulse spreading and thereby in-
troduces the inter-symbol interference in a received signal. Depending on the sign of the
GVD parameter either pulse spreading or pulse contraction may take place.The success of
high bit rate transmission through a standard fiber over a long distance depends on various
dispersion-compensating schemes.
Effect of higher order dispersion
Although the contribution of GVD dominates in most cases, at higher bit rates it is neces-
sary to consider the higher order dispersion. If the pulse wavelength nearly coincides with
zero dispersion wavelength, then decides the dominant contribution to GVD.
2.1.3 Polarization mode dispersion
Once dispersion is managed, polarization mode dispersion(PMD) becomes the most dom-
inant linear effect to limit the channel capacity. Single mode fibers actually support two
modes of polarization, due to the residual anisotropy produced by core ellipticity or non-
circularly symmetric stresses. The difference in the propagation constants of these two
principal states of polarization (PSP) gives rise to a differential transit time (differential
group delay (DGD)) for the propagated data stream. This causes a pulse broadening of
the output data, degrading the transmission performance.
2.2 Nonlinear impairments
When the optical communication systems operated at higher bitrates such as 10 Gbps
and above and/or at higher transmitter powers, it is important to consider the effects of
nonlinearities. In the case of SDM systems, nonlinear effects can become important even
at moderate powers and bit rates. The nonlinear effects that we consider in this section
6

17. arise owing to the dependence of the refractive index on the intensity of the applied electric
field, which in turn is proportional to the square of the field amplitude. At sufficiently
high optical intensities, nonlinear refraction occurs in the core (Kerr effect), which is the
variation of the index of refraction with light intensity. This makes Nonlinear Impairments
a critical concern in optical networks since long-haul transmission commonly relies on high
power lasers to transmit optical pulses over long spans to overcome attenuation. Nonlinear
Impairments depend mainly on the fiber type and length and can be placed into two
categories. The first includes the nonlinear effects that affect the energy of and optical
pulse and includes
• Stimulated Raman Scattering (SRS)
• Stimulated Brillouin Scattering (SBS)
• Four wave mixing (FWM)
Nonlinear effects that affect the shape of an optical pulse include:
• Self Phase Modulation (SPM)
• Cross Phase Modulation (XPM)
2.2.1 Stimulated Raman Scattering (SRS)
If two or more signals at different wavelengths are injected into a fiber, SRS causes power
to be transmitted from lower wavelength channels to the higher wavelength channels. The
incident light interacts with the molecular vibrations in a fiber and scattered light is gen-
erated, down-shifted by the Stokes frequency. If two optical waves separated by the Stokes
frequency are co-injected into a Ramanactive medium, the lower frequency (probe) wave
will experience optical gain generated by, and at the expense of the higher frequency (pump)
wave. This gain process is called thestimulated Raman scattering SRS. The magnitude of
the peak gain coefficient scales inversely with the pump wavelength. For a single injected
7

18. channel, the amplification of spontaneous Raman scattered light can cause depletion of the
signal light. In a multichannel system, SRS will couple channels separated by less than
about 110 nm, with power transfer from shorter to longer wavelengths.
2.2.2 Stimulated Brillouin Scattering (SBS)
SBS is a narrow band effect relative to the data channels operating in the terahertz
range resulting in shorter wavelengths amplifying longer wavelengths by depleting them-
selves.stimulated Brillouin scattering (SBS)involves the interaction of incident light with
acoustic waves in the silica glass generating down-shifted scattered light in a manner similar
to Stimulated Raman Scattering( presented above).
2.2.3 Four-wave mixing (FWM)
In fiber optical systems, using the angular frequencies ω1, ω2,...... ,ωN the intensity depen-
dence of the refractive index not only induces phase shifts within a channel but also gives
rise to signals at new frequencies such as 2ωiωj and ωi + ωjωk. This phenomenon is called
four-wave mixing. In contrast to SPM and XPM, which are significant mainly for high
bitrate systems, the four-wave mixing effect is independent of the bit rate but is critically
dependent on the channel spacing and fiber dispersion. Decreasing the channel spacing
increases the four-wave mixing effect, and so does decreasing the dispersion. In general the
signals generated by fourwave mixing have lower powers due to the lack of perfect phase
matching and the attenuation of signals due to fiber loss. The frequency components thus
generated are known as FWM products. If these fourwave Mixing products happen to
coincide with a signal channel, the interference causes a distortion of the signal amplitude.
As many channels of the system generate FWM products at the frequency of the distorted
channel, the interference can be regarded as random.
8

19. 2.2.4 Self- phase modulation (SPM)
In a single mode fiber, even a single light wave can be affected by this type of nonlinearity
since its phase is modulated by optical intensity fluctuations in the same wave. This effect
is called self-Phase Modulation (SPM). SPM together with GVD effect make the different
spectral components of optical pulse propagate at different speed. This make the pulse
broaden temporally and therefore leads to overlapping between adjacent bits and resulting
in an increase in bit error rate.
2.2.5 Cross phase modulation (XPM)
The innovation of space division multiplexing (SDM) enables multiple signals to be trans-
mitted on the same fiber. In a SDM system the optical intensity or power fluctuations
of a optical wave propagating in an optical fiber modulates the phase of the other co-
propagating optical signals through a phenomenon called Cross phase modulation (XPM).
XPM arises from the same phenomenon as SPM, but the effect of XPM is more relevant for
multi channel transmission. XPM can impose more limitations than SPM for transmission
systems since there are presumably many other channels to generate the phase shift and
intensity fluctuations.
9

20. 10

21. Chapter 3
Basic fiber optic communication
system
Fiber optics is a medium for carrying information from one point to another in the form of
light.Unlike the copper-wire form of transmission, fiber optics is not electrical in nature. A
basic fiber optic system consists of a transmitting device that converts an electrical signal
into a light signal, an optical fiber cable that carries the light, and a receiver that accepts
the light signal and converts it back into an electrical signal.
Fig. 3.1 Basic fiber optic communication system
11

22. The different elements of the system will vary according to the application. Systems
used for lower capacity links will employ somewhat different techniques and components to
those used by network providers that provide extremely high data rates over long distances.
Nevertheless the basic principles are the same whatever the system.
In the system the transmitter of light source generates a light stream modulated to
enable it to carry the data.The most commonly used optical transmitters are semiconductor
devices such as light-emitting diodes (LEDs) and laser diodes. The difference between
LEDs and laser diodes is that LEDs produce incoherent light, while laser diodes produce
coherent light. For use in optical communications, semiconductor optical transmitters must
be designed to be compact, efficient, and reliable, while operating in an optimal wavelength
range, and directly modulated at high frequencies.
The main component of an optical receiver is a photodetector, which converts light
into electricity using the photoelectric effect. The primary photodetectors for telecom-
munications are made from Indium gallium arsenide The photodetector is typically a
semiconductor-based photodiode. Several types of photodiodes include p-n photodiodes,
p-i-n photodiodes, and avalanche photodiodes.
12

23. Chapter 4
Modelling and implementation details
Like all other communication system, the primary objective of optical fiber communica-
tion system also is to transfer the signal containing information (voice, data, and video)
from the source to the destination. The source provides information in the form of electrical
signal to the transmitter. The electrical stage of the transmitter drives an optical source to
produce modulated light wave carrier. Semiconductor LASERs or LEDs are usually used
as optical source here. The information carrying light wave then passes through the trans-
mission medium i.e. optical fiber cables in this system. Now it reaches to the receiver stage
where the optical detector demodulates the optical carrier and gives an electrical output
signal to the electrical stage. The common types of optical detectors used are photodiodes
(p-i-n, avalanche), phototransistors, photoconductors etc. Finally the electrical stage gets
the real information back and gives it to the concerned destination.
In the area of optical fiber communication system design, it is crucial to enhance the com-
putational efficiency of waveform level simulation [1] of optical signal propagation through
the single mode fiber (SMF) [2]. The newly emerged coherent optical fiber communication
systems usually require simulation of long runs of symbols up to the order of one million to
directly estimate system performance with direct bit error counting due to nonlinear signal
clipping and quantization devices used in the coherent receiver [3, 4]. Furthermore, optical
13

24. fiber is a nonlinear channel and optical signals experience complex nonlinearity interaction
with both chromatic dispersion (CD) and polarization mode dispersion (PMD) [5, 6]. The
need for simulating long waveforms represents a huge computational burden. To handle
the complex interplay between fiber Kerr split-step Fourier (SSF) based methods are often
used [2]. By dividing a long optical fiber into small steps, the linear and nonlinear effects
are assumed to act separately during each step. In general, smaller simulation step-size
leads to higher simulation accuracy at a cost of higher computational burden. The SSF
based methods are computationally efficient due to the use of Fast Fourier transform (FFT)
to calculate the fiber chromatic dispersion effect.
4.1 Split-step Fourier (SSF) method
• The entire channel is divided into statistically independent sections where each section
may represent an optical device or a span of a few-mode fiber transmission line.
• The channel propagation matrix of each section represents the channel characteristics
of its corresponding physical component.
• When the number of modes is equal to D, the channel propagation matrix of the
kth
(1≤k≤K ) section represented Mk
(ω) is given by
Mk
(ω) = V k
Λk
(ω)(Uk
)∗
Where ∗ denotes Hermitian transpose. (Uk
) and (V k
) are frequency independent unitary
matrices representing mode coupling at the input and output of the kth
section respectively
and Λk
(ω) is a diagonal matrix with well-defined GDs having tauk as GD parameter, as
shown below
A =






e−jωτ1
0 . . .
...
...
0 e−jωτD






14

25. Fig. 4.1 (1)Matrix model for an MMF system described by SSF method in
cascade.(2) Each matrix in the cascade may represent a section of MMF, a
modal multiplexer or demultiplexer, a multimode optical amplifier or other
components.
Fig. 4.2 Matrix model for an MMF system described by the product of a
cascade of matrices
Where hij(t) represented in the figure is the time domain channel impulse response and
MN (i, j) is the channel matrix in frequency domain
4.2 Modelling of non-linearity in few mode fiber
4.2.1 Modelling of Kerr nonlinear effects
The Kerr effects were modelled by deriving the nonlinear pulse propagation equation for
a multimode fiber, starting by writing the induced nonlinear polarization as a function
15

26. of the electric field expansion into the two orthogonal polarization components of the N
orthogonal modes supported by the fiber.
The generalized coupled nonlinear Schrdinger equation is given by:
myequation
∂Aµi
∂z
+ β1µi
∂Aµi
∂t
−
jβ2µi
2
∂2Aµi
∂2t
+
αµi
2
Aµi (4.1)
= −j γµµii |Aµi|2
+ 2 v(v=µ) γµµii |Avi|2
+2
3 v γµvij |Avj|2
Aµi
Where i and j are the orthogonal states of polarization of each mode µ. Aµi(z, t),
β1µi , β2µi and αµi are the slowly varying field envelope, group velocity parameter, group
velocity dispersion parameter and attenuation parameter for the i polarization of the µ
mode, respectively. γµvij is the nonlinear coupling parameter between the i polarization of
mode µ and the j polarization of mode v, which depends on the nonlinear refractive index
n2 of the silica.
The first term of the right side is responsible for modal self-phase modulation (mSPM) of
the polarization i of the mode µ. The second term results in modal cross phase modulation
(mXPM) from the same polarization (i ) of different waveguide modes (v = µ). The third
term results also in mXPM, but coming from the orthogonal polarization ( j ) of the same
(v = µ) or different waveguide modes (v = µ).
16

27. 4.3 Space division multiplexing in few mode fiber
Space-division multiplexing (SDM) has emerged as a next-generation technology to sustain
the continuous traffic growth, in order to keep up with the future of Internet bandwidth
requirements. Among several techniques, mode-division multiplexing (MDM) transmission
systems utilizing few-mode fibers (FMF) have been intensively explored. In MDM systems,
each data channel is modulated onto individual spatial or polarization mode to increase
the overall number of parallel channels, thus enabling higher transmission capacity.
Fig. 4.3 Space division multiplexing(SDM) Using few mode fiber with DSP
MIMO processing unit
Data is transmitted using different modes which are launched into the optical fiber
channel using a spatial mode multiplexer.In its path we can use an Optical Add Drop
Multiplexer(OADM) for signal add or drop purpose according to the requirement. Then in
the receiver side we use a spatial mode demultiplexer to separate the modes into individual
modes.As there will be coupling and inter-modal crosstalk between the modes during the
transmission we will not be able to recover the same data that we have sent in respective
modes.To overcome this we will take the help of DSP-MIMO Processor to estimate the
collected data correctly.
17

28. 4.4 MIMO signal processing in few mode fiber for SDM
One of the fundamental challenges in an FMF transmission systems is the inter-modal
crosstalk between any two spatial or polarization modes. Another significant challenge
is the large accumulated differential mode group delay (DMGD), which makes mode de-
multiplexers even more difficult to be implemented in either optical or electrical domain.
It has been proposed and demonstrated that adaptive multi-input multi-output (MIMO)
equalization can dynamically compensate DMGD and demultiplex the signals on different
modes using digital signal processing (DSP). Among several adaptive MIMO equalization
methods, least mean square (LMS) algorithm is considered as an attractive approach in
the FMF systems, because it is a good compromise among equalization performance, hard-
ware complexity, and dynamic speed [10], [11]. The adaptive LMS algorithm can be either
implemented in time domain or frequency domain. Compared with the time domain ap-
proach, the adaptive frequency domain method has been proven to be much more hardware
efficient in FMF transmission systems. Several experimental results have also confirmed
better hardware efficiency of the frequency domain approach.
In addition to hardware complexity, the convergence speed of the adaptive MIMO equal-
izer is another key consideration, which may significantly impact the system performance.
In a training sequence based frequency domain LMS (FD-LMS) equalizer, training symbols
are used for initial channel estimation, and then, a decision-directed adaptive method is
used for continuous channel adaptation. In such an equalizer architecture, slower conver-
gence speed of the adaptive FD-LMS method may require a longer training sequence in
the system for its initial channel estimation, thus decreasing the overall system efficiency.
In addition, the MIMO equalizer should also be able to track the fast-changing channels of
FMF transmission systems during initial channel estimation. .
18

29. Chapter 5
Results
Fig. 5.1 Scatter plot corresponding to mode1 in a two mode system when
input power =0dbm for both the signal entering into different modes not
considering nonlinearity in FMF.
19

30. Fig. 5.2 Scatter plot corresponding to mode1 in a two mode system when in-
put power =0dbm for both the signal entering into different modes considering
nonlinearity in FMF.
Fig. 5.3 Scatter plot corresponding to mode2 in a two mode system when
input power =0dbm for both the signal entering into different modes and not
considering nonlinearity in FMF.
20

31. Fig. 5.4 Scatter plot corresponding to mode2 in a two mode system when
input power =0dbm for both the signal entering into different modes and
considering nonlinearity in FMF.
Fig. 5.5 Scatter plot corresponding to mode1 in the first plot and mode2
in the second plot in a two mode system when peak input power =-5dbm for
both the signal entering into different modes and not considering nonlinearity
in FMF
21

32. Fig. 5.6 Scatter plot corresponding to mode1 in the first plot and mode2
in the second plot in a two mode system when peak input power =-5dbm for
both the signal entering into different modes and considering nonlinearity in
FMF.
Fig. 5.7 Scatter plot corresponding to mode1 in the first plot and mode2
in the second plot in a two mode system when peak input power = 5dbm for
both the signal entering into different modes and not considering nonlinearity
in FMF.
22

33. Fig. 5.8 Scatter plot corresponding to mode1 in the first plot and mode2
in the second plot in a two mode system when peak input power = 5dbm for
both the signal entering into different modes and considering nonlinearity in
FMF
Fig. 5.9 Scatter plot corresponding to mode1 in the first plot and mode2
in the second plot in a two mode system when peak input power for mode1=
7dbm for mode1 and peak input power for mode2= 3dbm for the signal en-
tering into different modes and not considering nonlinearity in FMF.
23

34. Fig. 5.10 Scatter plot corresponding to mode1 in the first plot and mode2
in the second plot in a two mode system when peak input power for mode1=
7dbm for mode1 and peak input power for mode2= 3dbm for the signal en-
tering into different modes and considering nonlinearity in FMF
24

35. Chapter 6
Conclusion and discussion
We have studied the performance of few mode fiber in different situations. We also see
the symmetric split-step Fourier method for solving the nonlinear Schrdinger equation ap-
plied to various optical fiber transmission systems. When implementing a step-size control
method, one must be aware of the following issues. First, all the estimates for the local
error have been derived for a step-size hk in a neighborhood of 0 and they are used for
step size control where the goal is to determine step-size as large as possible to reduce the
computational cost. It is one of the reason why a good step-size control strategy must
include safeguard so that the step-size does not increase in an inconsiderate way.
Our aim was to implement a system comprising of few mode fiber which will use Space
Division Multiplexing technique and MIMO processing to establish a communication using
fiber optics. For this purpose we have designed a simulation model of a few mode fiber
acting as the channel for the communication. We have taken into account the non-linearity
associated with the few mode fiber so that the model becomes more realistic in its ap-
proach.Taking into consideration of non-linearity is very much imortant as in few mode
fiber the chances of inter-modal cross talk is very high. The data transmission in few mode
fiber becomes more and more complicated due to mode coupling and other non-linearity
phenomenon that occurs in few mode fiber. As we can see from our results when we take
25

36. two modes and send signal of same power in them their behavior is almost same in both
the modes as seen at the output side. But when we take non linearity into account
We have successfully designed the model to simulate few mode fiber.The model can
be used to simulate a given few mode fiber provided its non- linearity parameters such
as attunation parameter,coupling parameter etc are known to us.If we do the analysis of
the result that we are obtaining it can be shown that as the input power of the signal
increases the non-linearity effects came into existence more prominently.As we can see
from the results (Fig (5.2),Fig(5.5),Fig(5.8)) when we give power 0dbm,-5dbm and 5 dbm
respectively to both the signals travelling through two different modes in the fiber,the signal
corresponding to the 5dbm input power is highly distorted where signal corresponding to
peak power -5dbm is least distorted and the distortion level of the signal whose power
corresponding to 0 dbm remains in between the -5dbm and 5 dbm signal.
Also we can see that(from Fig 5.10 and Fig(5.8))if we keep the power of the launched
signal same for both the modes then their variation in the scatter plot is almost same where
as if we give different peak power to the signals launched into them then the scatter plot
will no longer be equivalent for both of the modes.It will vary considerably.It can also be
concluded from the results that when we don’t take non-linearity into consideration the
scatter plot does not vary much.But if we take into account the non-linearity effect into
consideration the both the graphs have considerable amount of changes between them.So
for realistic and practical problems it is always desirable to take into account the non-
linearity effects of the few mode fiber.Otherwise it will cost us huge amount of data loss
and lots of complexities in ous designed systems.
There remain tremendous challenges to be solved before one can truly assess the value of
spatial multiplexing in fibers for high-capacity commercial systems. SDM offers the ability
to take advantage of the highest degree of optical and electronic integration, a technological
path that often led to lower system cost when integrated components can be mass produced
and is not performed at the expense of overall performance. Even though the deployment
26

37. of new optical fibers is necessary before high-capacity space-division multiplexed systems
operating over a single fiber strand become a commercial reality, ultimately, it is whether
one can achieve a lower cost per bit transported that determines commercial viability. The
intense activity in the area of SDM foreseen in the next few years should provide some
answers on the future of this technology.
27

38. 28

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30